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Text By Zeckendorf's theorem each positive integer is uniquely written as a sum of distinct non-adjacent terms of the Fibonacci sequence. This representability remains true for so called the Nth order Fibonacci sequence , and for a further generalization to linear recurrences with positive coefficients. In this paper we consider sequences { G n } that have the same linear recurrence relations as the N th order Fibonacci sequence but has different initial values, and investigate the number of positive integers up to X that are written as a sum of distinct terms of G n . We also introduce a converse of Zeckendorf's theorem that does not require the increasing condition. Our method extends to general linear recurrences, and a generalization is introduced in this paper. Video For a video summary of this paper, please visit https://youtu.be/vSwSJ_sppns . [ABSTRACT FROM AUTHOR]

Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

Abstract The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers. [ABSTRACT FROM AUTHOR]

Abstract: Given a set let denote the number of ordered pairs such that . The celebrated Erdős–Turán conjecture states that if such that for all sufficiently large n, then the representation function must be unbounded. For each positive integer m, let be the least positive integer r such that there exists a set with and . Ruzsa''s method in [I.Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990) 145–151] implies that must be bounded. It is pleasure to call a Ruzsa''s number. In this paper we prove that all Ruzsa''s numbers . This improves the previous bound . Several related open problems are proposed. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=hgDwkwg_LzY. [Copyright &y& Elsevier]

Let p be a prime number, q = p r with r ≥ 2 and P ∈ F q [ X ] . In this paper, we first estimate the number of x ∈ F q such that P ( x ) has prescribed digits (in the sense of Dartyge and Sárközy). In particular, for a given proportion <0.5 of prescribed digits, we show that this number is asymptotically as expected. Then, we obtain similar results when x is allowed to run only in the set of generators (primitive elements) of F q ⁎ . In the case of special interest where P is a monomial of degree 2, our estimate for the number of x ∈ F q such that P ( x ) has prescribed digits is sharper than the estimate following from the Weil bound. We will need to study exponential sums of independent interest such as multiplicative character sums over affine subspaces and additive character sums with generator arguments. [ABSTRACT FROM AUTHOR]

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B . B : = { a b : a , b ∈ B } . In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers. [ABSTRACT FROM AUTHOR]

A univariate polynomial f ( x ) is said to be nice if all of its coefficients as well as all of the roots of both f ( x ) and its derivative f ′ ( x ) are integers. The known examples of nice polynomials with distinct roots are limited to quadratic polynomials, cubic polynomials, symmetric quartic polynomials and, up to equivalence, only a finite number of nonsymmetric quartic polynomials and one quintic polynomial. In this paper we find parametrized families of nice nonsymmetric quartic polynomials with distinct roots, as well as infinitely many nice quintic and sextic polynomials with distinct roots. [ABSTRACT FROM AUTHOR]

Abstract: We say the sets of nonnegative integers and are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set by using analytic tools. [Copyright &y& Elsevier]

Abstract: Let be positive integers. In this paper, we prove that the equation has no solutions in integers and k with , and . [Copyright &y& Elsevier]

Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]

Abstract: Let m be a positive integer and be a polynomial of the form . We call a polynomial a Rabinowitsch polynomial if for and consecutive integers , is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials . [Copyright &y& Elsevier]

Abstract: Let n be a fixed odd integer with . In this paper, using a recent result on the existence of primitive divisors of Lehmer numbers give by Y. Bilu, G. Hanrot and P.M. Voutier, we prove that the equation has no positive integer solution with . [Copyright &y& Elsevier]

Abstract: Let denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that any sufficiently large integer N satisfying the congruence condition can be represented as the sum of twelve fourth powers of primes and the fourth power of one . This result constitutes an improvement upon that of Ren and Tsang. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and be the smallest positive integer s such that every integer is a sum of s kth powers . We establish and provided that k is not divisible by . Next, let , and q be any positive integer. We show that if then for some constant . These results generalize results known for the case of prime moduli. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0. [Copyright &y& Elsevier]

In 1934, Romanoff proved that there are a positive proportion natural numbers which can be expressed as the sum of a prime and a power of 2. In this paper, a quantitative version of this theorem is given. We show that the proportion is larger than 0.0868 and for a positive proportion of odd integers the number of such representations is between 1 and 16. [Copyright &y& Elsevier]